x-R-positive-real-numbers-Prove-that-x-2-2-x-3-

Question Number 58600 by naka3546 last updated on 26/Apr/19

x \in R^{+} (p o s i t i v e r e a l n u m b e r s)

P r o v e t h a t

x^{2} + \frac{2}{x} ⩾ 3

Answered by tanmay last updated on 26/Apr/19

x^{3} - 3 x + 2

x^{3} - 3 x + 3 - 1

x^{3} - 1 - 3 (x - 1)

(x - 1) (x^{2} + x + 1) - 3 (x - 1)

(x - 1) (x^{2} + x + 1 - 3)

(x - 1) (x^{2} + 2 x - x - 2)

(x - 1) {x (x + 2) - 1 (x + 2)}

(x + 2) {(x - 1)}^{2}

(x + 2) > 0 a n d {(x - 1)}^{2} ⩾ 0 w h e n x = 1

s o x^{3} - 3 x + 2 ⩾ 0

x^{2} - 3 + \frac{2}{x} ⩾ 0

x^{2} + \frac{2}{x} ⩾ 3

Answered by salahahmed last updated on 26/Apr/19